Abstract: Seismic anisotropy has been extensively acknowledged as a crucial element that influences the wave propagation characteristic during wavefield simulation, inversion and imaging. Transversely isotropy (TI) and orthorhombic anisotropy (OA) are two typical categories of anisotropic media in exploration geophysics. In comparison of the elastic wave equations in both TI and OA media, pseudo-acoustic wave equations (PWEs) based on the acoustic assumption can markedly reduce computational cost and complexity. However, the presently available PWEs may experience SV-wave contamination and instability when anisotropic parameters cannot satisfy the approximated condition. Exploiting pure-mode wave equations can effectively resolve the above-mentioned issues and generate pure P-wave events without any artifacts. To further improve the computational accuracy and efficiency, we develop two novel pure qP-wave equations (PPEs) and illustrate the corresponding numerical solutions in the time-space domain for 3D tilted TI (TTI) and tilted OA (TOA) media. First, the rational polynomials are adopted to estimate the exact pure qP-wave dispersion relations, which contain complicated pseudo-differential operators with irrational forms. The polynomial coefficients are produced by applying a linear optimization algorithm to minimize the objective function difference between the expansion formula and the exact one. Then, the developed optimized PPEs are efficiently implemented using the finite-difference (FD) method in the time-space domain by introducing a scalar operator, which can help avoid the problem of spectral-based algorithms and other calculation burdens. Structures of the new equations are concise and corresponding implementation processes are straightforward. Phase velocity analyses indicate that our proposed optimized equations can lead to reliable approximation results. 3D synthetic examples demonstrate that our proposed FD-based PPEs can produce accurate and stable P-wave responses, and effectively describe the wavefield features in complicated TTI and TOA media.